Bochner's formula

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In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold $(M, g)$ to the Ricci curvature. More specifically, if $u : (M, g) \rightarrow \mathbb{R}$ is a harmonic function, so $\triangle_g u = 0$ ( $\triangle$ is the Laplacian operator), then $\triangle \frac{1}{2}|\nabla u| ^2 = |\nabla^2 u|^2 - \mbox{Ric}(\nabla u, \nabla u)$ . The formula is an example of a Weitzenböck identity. Bochner used this formula to prove the Bochner vanishing theorem.